Revisiting The Basic Physics Of Radioactivity

 Radioactivity basically occurs in three forms, as determined by the particles: alpha-radiation (from alpha particles or Helium nuclei), beta radiation, from beta particles or electrons, and gamma radiation, from gamma particles or very high energy photons (e.g. at very short wavelengths, hence can easily penetrate tissue). The simple diagram below graphically shows the differences in the radiations with respect to an applied magnetic field. 

If one holds the thumb of one's RIGHT hand into the image (to represent the B-field direction) then the electrons (beta particles) will display a direction coincident with the curving fingers of the right hand. That is, clockwise. Now, since the alpha particles are positively charged (, e.g. He⁺⁺, as opposed to the negatively charged beta particles, e.g. e-) they will go in the opposite direction. 

The gamma rays, meanwhile, suffer no deflection in the field because they have zero charge, being photons of light. In terms of penetration power these additional differences apply:

1) Alpha ( a ) particles are absorbed by a few cm of air, or by an aluminum foil only 0.006 cm thick

2) Beta (b)  particles - while having less ionizing power than alpha particles (because of much lower mass) have 100 times more penetrating power. A sheet of aluminum at least 3mm thick is needed to absorb them.

3) Gamma rays produce little ionization since they have no electric charge but can pass through a block of iron a foot thick.

The Activity  of a radioactive source. 

We define the activity of a radioactive source as:

A = dN/dt = - lN

 

Where  l  is the decay constant. The negative sign appended to the equation indicates that the amount N is decreasing with time t.

The units are shown in the relationship below:

A [Bq]  = dN/dt [s^-1]  = - lN[s^-1]

Where Bq is Becquerels.

 

The decay curve is obtained from the fundamental law of radioactive decay, based on some original number of atoms N_o decaying with an activity l over time t:

 

N =  N_o  exp (-lt)

 

Then:  ½d N/dt ½  = N_o  l exp (-lt)= R

 

Where  R is the decay rate, i.e. R = R_o  exp (-lt)

 

And:  R_o  = N_o  l is the decay rate at time t = 0.

The half-life is the time for half of the original (N_o ) atoms to disintegrate or when the point is reached such that:

 

N ®  N_o  /2  or  R ®  R_o  /2 

 

 Then:    N_o  /2  =  N_o  exp (-l T _½)

 

 Where  T _½ is specifically substituted for t.

 

After dividing N_o into both sides and taking natural logarithms we get:

 

l T_½  =  ln 2 = 0.693

 

Or:   T _½  =  ln 2/ l  =   0.693/ l

 

Using this basis any sample or fossil with even a minuscule amount of radioactive material can be dated. All we need know is that over the period defined as T½  half of the number of the remaining atoms decay and the activity is in Becquerels (Bq).  Thus, if  T_½ = 15,000 yrs.  for  l = 200 Bq then if l  = 50 Bq now the sample is 45,000 years old. A graphical depiction of generic radionuclide decay is shown below :

It is easy to see from the graph shown that the half-life of this nuclide is about 1 million years, so: T½ =  10⁶ yrs.   The problem of dating radioactive fossils or other specimens  (e.g. the cloth of Turin) is really a problem of finding a radioactive isotope that is most appropriate, or one that enables the maximum accuracy for the time scale desired.  Hence, for ancient fossils one would look for isotopes that have half-lives in the millions rather than thousands of years. Failing that one would wish to have available some kind of correction method, say to correct for extraneous effects such as the atmosphere might impose on samples.

In many ordinary fossil dating applications, potassium-argon methods are employed, based on the relative compositions of Potassium -40 to Argon-40 gas. Typically when rocks or other items are tested the sample is split between the Potassium-40 content on the one hand and the Argon on the other. The instrument of choice to assess the ratio: K40/Ar 40 is the mass spectrometer.

 

 Exotic isotopes of carbon can also be used is the measurement technique is sufficiently refined. In a recent use of the isotope d ¹³C, for instance, evidence has been found for the existence of life on Earth at least 3,850 million years ago. For this purpose, quartz (zircon, zirconium) crystals have been found to be of use since they may harbor small amounts of thorium and uranium at the level of parts per billion.

For dating samples in the millions of years, particularly for igneous rocks and samples embedded within, isotopes of lead and strontium may be used – being the ‘daughters’ from millions of years of radioactive decay.

 

Problems: 

1) A point source of gamma radiation has (T½) = 30 mins. The initial count rate recorded by a G-M tube is 360/s. Find the count rate that would be recorded after 4 half lives. Sketch the decay curve and determine the activity, A.


2)  Find the half life of the beta particle emitting nuclide: 

 ³²P ₁₅,

If the activity A = 5.6 x 10^-7 /s.

3) A radionuclide sample of N = 10¹⁵ atoms undergoes decay at the constant average rate of dN/dt =  6.00 x  10¹¹  /s.  

From this information, find:

a) The Activity A

b) The decay constant l

c) The half life of the sample in minutes.

4) The activity of a radio-nuclide is given as:

A =  A_o  exp (-lt)

Where  A_o is the decay rate at time t = 0, and A refers to the decay rate at some time t thereafter. If a particular radio-nuclide has  A_o  = 1. 1 x 10 ¹⁰  decays/sec and a half life T_1/2 = 28.0 years, find:

a) The decay constant, l ,

b) The activity  A after 1 hour,  after 2 hours.

c) The activity  A  after 49 years,